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Topic: Matrix multiplication

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  Matrix (mathematics) - Wikipedia, the free encyclopedia
The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A.
A 1 × n matrix (one row and n columns) is called a row vector, and an m × 1 matrix (one column and m rows) is called a column vector.
The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.
en.wikipedia.org /wiki/Matrix_(mathematics)   (1664 words)

 Matrix multiplication - Wikipedia, the free encyclopedia
Matrix multiplication is not commutative (that is, AB ≠ BA), except in special cases.
Although matrix multiplication is not commutative, the determinants of AB and BA are always equal (if A and B are square matrices of the same size).
Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication", is based on a clever way of multiplying two 2 × 2 matrices which requires only 7 multiplications (instead of the usual 8).
en.wikipedia.org /wiki/Matrix_multiplication   (1036 words)

 Encyclopedia :: encyclopedia : Multiplication   (Site not responding. Last check: 2007-10-22)
Frequently, multiplication is implied by Juxtaposition rather than shown in a notation.
Once multiplication has been defined for natural numbers, it can be extended to include integers, and then to real and complex numbers.
Most, such as lattice multiplication, require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9); the peasant multiplication algorithm does not.
www.hallencyclopedia.com /Multiplication   (841 words)

 PlanetMath: matrix operations
If a matrix has exactly as many rows as it has columns, we say it is a square matrix.
Multiplication of two matrices is allowed provided that the number of columns of the first matrix equals the number of rows of the second matrix.
Under the correspondence of matrices and linear transformations, one can show that matrix multiplication is equivalent to composition of linear transformations, which explains why matrix multiplication is defined in a manner which is so odd at first sight, and why this strange manner of multiplication is so useful in mathematics.
planetmath.org /encyclopedia/MatrixOperations.html   (459 words)

 SMM: Matrices and Matrix Multiplication; spreadsheet models
A matrix that has the same number of rows as columns is said to be "square." The (3,2) element of a matrix is the number that is in the second column of the third row.
The transpose of a matrix is a new matrix that is obtained by interchanging the rows and columns of the original matrix.
The notation for the transpose of a matrix A is A
www.chacocanyon.com /smm/readings/matrix.shtml   (1217 words)

 Matrix Multiplication
Discussion: Although matrix multiplication is an important problem in linear algebra, its main significance for combinatorial algorithms   is its equivalence to a variety of other problems, such as transitive closure and reduction, solving linear systems, and matrix inversion.
Matrix multiplication arises in its own right in computing the results of such coordinate transformations as scaling, rotation, and translation for robotics and computer graphics.
Matrix multiplication has a particularly interesting interpretation in counting the number of paths between two vertices in a graph.
www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK3/NODE138.HTM   (910 words)

 Matrix Algebra
Elements (entries) of the matrix are referred to by the name of the matrix in lower case with a given row and column (again, row comes first).
(The main or principal diagonal in matrix B is composed of elements all equal to 1.) With a square, symmetric matrix, the transpose of the matrix is the original matrix.
Multiplying a matrix by the identity matrix is analogous to the real number operation of multiplying a number or variable by 1: the resulting output is identical to the numbers input.
luna.cas.usf.edu /~mbrannic/files/regression/matalg.html   (1775 words)

 SparkNotes: Matrices: Matrix Multiplication
To multiply a matrix by a scalar, that is, a single constant, variable, or expression, multiply all the entries in the matrix by the scalar:
The answer will be a matrix with the same number of rows as the first matrix and the same number of columns as the second matrix.
Matrix multiplication is not necessarily commutative: it is not always true that AB = BA.
www.sparknotes.com /math/algebra2/matrices/section2.rhtml   (464 words)

 math lessons - Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure.
A matrix with m rows and n columns is called an m-by-n matrix (or m×n matrix) and m and n are called its dimensions.
Multiplication of two matrices is well-defined only if the number of columns of the first matrix is the same as the number of rows of the second matrix.
www.mathdaily.com /lessons/Matrix_(mathematics)   (1494 words)

 Math Forum - Ask Dr. Math
In any case, a matrix is a rectangular array of numbers, as you probably know, with the numbers arranged in rows and columns.
A matrix with the same number of rows and columns is called a square matrix.
Matrix multiplication is defined in a rather peculiar fashion.
mathforum.org /library/drmath/view/51458.html   (816 words)

One of the most useful properties of the matrix is the matrix multiplication.
= the inverse matrix of a, and I = the identity matrix.
For a square matrix to be invertible, its determinant must not be 0.
kwon3d.com /theory/vect/matrix.html   (456 words)

 Matrix Multiplication
Matrix multiplication (also called dot or inner product) is carried out in Mathematica with the function Dot, typically entered with a dot short-hand syntax.
This demonstrates matrix multiplication of a matrix with itself.
Matrix multiplication is a fundamental operation of linear algebra computation.
documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/2.7.html   (283 words)

 Tuning Strassen's Matrix Multiplication for Memory Efficiency
Strassen's algorithm for matrix multiplication gains its lower arithmetic complexity at the expense of reduced locality of reference, which makes it challenging to implement the algorithm efficiently on a modern machine with a hierarchical memory system.
23] for matrix multiplication and its variants are the most practical of such algorithms, and are classic examples of theoretically high-performance algorithms that are challenging to implement efficiently on modern high-end computers with deep memory hierarchies.
To understand this phenomenon, consider that for matrix sizes of 505 to 512 the padded matrix size is 512 and the recursion truncation point is at tile size 32.
www.cs.duke.edu /~alvy/papers/sc98/index.htm   (5218 words)

 Direct Matrix Multiplication   (Site not responding. Last check: 2007-10-22)
Matrix multiplication is one of the most basic mathematical and scientific algorithms.
The ``direct'' matrix multiplication algorithm is relatively easier and potentially more efficient since the operand arrays have carefully chosen replicated/collapsed distributions.
The reason is the innermost loop for the direct matrix multiplication algorithm in HPJava is ``for'' loop, i.e.
www.hpjava.org /theses/hkl/dissertation/dissertation/node67.html   (337 words)

 Construct: Matrix Class Reference   (Site not responding. Last check: 2007-10-22)
Matrix (const double m00, const double m01, const double m02, const double m03, const double m10, const double m11, const double m12, const double m13, const double m20, const double m21, const double m22, const double m23, const double m30, const double m31, const double m32, const double m33)
Creates a transformation matrix representing a scaling by the vector 's', the result's are stored in "this".
The j position in the matrix to retrieve the data from.
voronoi.sbp.ri.cmu.edu /software/Construct/doc/html/classMatrix.html   (368 words)

 Matrix Chain Multiplication
A Matrix Chain Multiplier is perhaps the quintessential example of dynamic programming, a technique that nearly every data structures and algorithms book explores.
You may input your own dimensions for a matrix chain or have the computer generate them for you.
If you choose to specify your own dimensions, you also have the option to input integer values for the matrix entries, since this program is capable of performing the multiplication.
www.brian-borowski.com /Matrix   (140 words)

 Matrix Operations
The result is a new matrix with the same dimensions in which each element is the sum of the corresponding elements of the previous matrices.
The inverse of a square matrix is a matrix of the same size that, when multiplied by the matrix, gives an identity matrix of the same size.
Multiplication by the inverse of a matrix is like dividing by the matrix, except this is strictly true only if the matrix is {1*1}.
www.stanford.edu /~wfsharpe/mia/mat/mia_mat2.htm   (1904 words)

 Interpretation of matrix multiplication?
Matrix multiplication is clearly defined but is there a tangible or physical interpretation for it?
I am thinking about each column of the matrix as vectors so matrix multiplication with two 2by2 matrices is about multiplying 4 vectors in a certain way.
A good interpretation is this: You can view your matrix as the matrix of a linear transformation (mapping) between two vector spaces (of finite dimension) in fixed basis for the two vector spaces.
www.physicsforums.com /showthread.php?p=1060534#post1060534   (645 words)

 Matrix Algebra for Markov Chains
Matrix algebra refers to computations that involve vectors (rows or columns of numbers) and matrices (tables of numbers), as wells as scalars (single numbers).
If A is a matrix and c is a number (sometimes called a scalar in this context), then the scalar multiple, cA, is obtained by multiplying every entry in A by c.
The inverse of a square matrix A is a matrix, often denoted by A
home.ubalt.edu /ntsbarsh/Business-stat/Matrix/Mat4.htm   (778 words)

 Matrix Multiplication   (Site not responding. Last check: 2007-10-22)
Multiplication -- from MathWorld The product \mathsf{C} of two matrices \mathsf{A} and \mathsf{B} is defined as c_{ik} = a_{ij}b_{jk}, where j is summed over Water Pressure Pump for all possible values of i and k and.
Since matrix multiplication is associative, the order in which multiplications are performed.
The program is simply a repeated matrix multiplication of two different matrices, with a third matrix.
matrix.ksearch.in /matrix-multiplication.htm   (541 words)

 Matrix multiplication: an interactive micro-course for beginners
Matrix multiplication is a very useful operation in mathematics, although the definition may seem a bit difficult and unnatural when first encountered.
First, here is how to multiply a row matrix with a column matrix of the same length - that is, with the same number of entries.
The resulting numbers are arranged in a new matrix: the m-th row in A times the n-th column in B gives the number at position (m,n) in AB:
www.mai.liu.se /~halun/matrix/matrix.html   (318 words)

 Matrix Chain Multiplication   (Site not responding. Last check: 2007-10-22)
Since matrix multiplication is associative, the order in which multiplications are performed is arbitrary.
For example, let A be a 50*10 matrix, B a 10*20 matrix and C a 20*5 matrix.
Your job is to write a program that determines the number of elementary multiplications needed for a given evaluation strategy.
acm.uva.es /p/v4/442.html   (314 words)

 Improving the memory-system performance of sparse-matrix vector multiplication
The bandwidth of a sparse matrix is the maximum distance, in diagonals, between two nonzero elements of the matrix.
The use of prefetching to prevent multiple load/store units from stalling on the same cache line is likely to have a beneficial effect on other RISC processors that have multiple load/store units.
This multiplication is done by an arithmetic shift instruction that executes in one cycle on one of the integer ALUs.
www.research.ibm.com /journal/rd/416/toledo.html   (7444 words)

 Multiplication of Matrices
In fact, the general rule says that in order to perform the multiplication AB, where A is a (mxn) matrix and B a (kxl) matrix, then we must have n=k.
The matrix multiplication is not commutative, the order in which matrices are multiplied is important.
More on the multiplication of matrices, may be found in the next page.
www.sosmath.com /matrix/matrix1/matrix1.html   (469 words)

 6.2 - Operations with Matrices
The number of columns in the first matrix must be equal to the number of rows in the second matrix.
Since the number of columns in the first matrix is equal to the number of rows in the second matrix, you can pair up entries.
Multiplication by the identity matrix is commutative, although the order of the identity may change
www.richland.edu /james/lecture/m116/matrices/operations.html   (833 words)

 Matrix   (Site not responding. Last check: 2007-10-22)
Five fundamental matrix decompositions, which consist of pairs or triples of matrices, permutation vectors, and the like, produce results in five decomposition classes.
These decompositions are accessed by the Matrix class to compute solutions of simultaneous linear equations, determinants, inverses and other matrix functions.
Note that is the matrix is to be read back in, you probably will want to use a NumberFormat that is set to US Locale.
math.nist.gov /javanumerics/jama/doc/Jama/Matrix.html   (919 words)

 MATH5315 - Matrix Multiplication   (Site not responding. Last check: 2007-10-22)
This is typically due to the use of operations which are less efficient in that particular application or on the order in whihc the operations are done.
The file matmul.m is a Matlab m-file to time the calculation of C = A*B using both the intrinsic Matlab matrix multiplication and the for loops as you would naievly implement the definition of matrix multiplication.
As matrix multiplication is a basic operation, highly efficient implementations are usually provided by the vendor as part of the BLAS library.
web.maths.unsw.edu.au /~rsw/MATH5315/matmul.shtml   (796 words)

 A three-dimensional approach to parallel matrix multiplication
Most parallel matrix multiplication algorithms used as building blocks in scientific applications are 2D algorithms.
In the complex case, there is an additional advantage, since it is possible to multiply two complex matrices together using three real matrix multiplications and five real matrix additions instead of four real matrix multiplications and two real matrix additions [1].
The new scheme for partitioning matrices across processors presented in conjunction with the 3D matrix multiplication algorithm is applicable to most of the level-3 BLAS.
www.research.ibm.com /journal/rd/395/agarwal.html   (2587 words)

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